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Cop-Width, Flip-Width and Strong Colouring Numbers.

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Title: Cop-Width, Flip-Width and Strong Colouring Numbers.
Authors: Hickingbotham, Robert1
Source: Discrete Mathematics & Theoretical Computer Science (DMTCS). 2025, Vol. 27 Issue 2, p1-9. 9p.
Subjects: Sparse graphs
Abstract: Cop-width and flip-width are new families of graph parameters introduced by Toruńczyk (2023) that generalise treewidth, degeneracy, generalised colouring numbers, clique-width and twin-width. In this paper, we bound the cop-width and flip-width of a graph by its strong colouring numbers. In particular, we show that for every r ∈ N, every graph G has copwidthr (G) ≤ scol4r(G). This implies that every class of graphs with linear strong colouring numbers has linear cop-width and linear flip-width. We use this result to deduce improved bounds for cop-width and flip-width for various sparse graph classes. [ABSTRACT FROM AUTHOR]
Copyright of Discrete Mathematics & Theoretical Computer Science (DMTCS) is the property of Discrete Mathematics & Theoretical Computer Science DMTCS and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: Cop-width and flip-width are new families of graph parameters introduced by Toruńczyk (2023) that generalise treewidth, degeneracy, generalised colouring numbers, clique-width and twin-width. In this paper, we bound the cop-width and flip-width of a graph by its strong colouring numbers. In particular, we show that for every r ∈ N, every graph G has copwidthr (G) ≤ scol4r(G). This implies that every class of graphs with linear strong colouring numbers has linear cop-width and linear flip-width. We use this result to deduce improved bounds for cop-width and flip-width for various sparse graph classes. [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of Discrete Mathematics & Theoretical Computer Science (DMTCS) is the property of Discrete Mathematics & Theoretical Computer Science DMTCS and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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      – TitleFull: Cop-Width, Flip-Width and Strong Colouring Numbers.
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              Text: 2025
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